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arXiv:2402.09502 · Physical Review Research

Deconfined quantum criticality of nodal d -wave superconductivity, Néel order, and charge order on the square lattice at half-filling

Maine Christos, Henry Shackleton, Subir Sachdev A, Zhu-Xi Luo

Published Jul 2, 2024• 10 citations
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Abstract

We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the π -flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background π -flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal d -wave superconductor, and states with Néel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases that are very likely described by ( 2 + 1 ) -dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and Néel order, and a conventional d -wave superconductor with gapless Bogoliubov quasiparticles at four nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of Néel, valence bond solid (Kekulé), and Dirac semimetal phases.

DOI
10.1103/physrevresearch.6.033018

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We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the π -flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background π -flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal d -wave superconductor, and states with Néel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases that are very likely described by ( 2 + 1 ) -dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and Néel order, and a conventional d -wave superconductor with gapless Bogoliubov quasiparticles at four nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of Néel, valence bond solid (Kekulé), and Dirac semimetal phases.
We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the hole-doped cuprates. The spin liquid can be described by a SU(2) gauge theory of N f = 2 massless Dirac fermions carrying fundamental gauge charges—this is the low-energy theory of a mean-field state of fermionic spinons moving on the square lattice with π -flux per plaquette in the ℤ 2 center of SU(2). This theory has an emergent SO(5) f global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion U at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in π ℤ 2 -flux. At half-filling, the low-energy theory of the Higgs sector has N b = 2 relativistic bosons with a possible emergent SO(5) b global symmetry describing rotations between a d -wave superconductor, period-2 charge stripes, and the time-reversal breaking “ d -density wave” state. We propose a conformal SU(2) gauge theory with N f = 2 fundamental fermions, N b = 2 fundamental bosons, and a SO(5) f ×SO(5) b global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5) f and a confining state which breaks SO(5) b . The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order and d -wave superconductivity. A similar theory applies at nonzero doping and large U , with longer-range couplings of the chargons leading to charge order with longer periods.
We present a quantum Monte Carlo study of the transition between the insulating Néel state and the nodal d -wave superconductor on the square lattice at half-filling. We access a regime of frustrated magnetic order without a sign problem using a parton representation of the electron in terms of fermionic spinons and bosonic chargons. Both partons move in a background π -flux (so the electron experiences no net flux) and are coupled to a quantum fluctuating SU(2) lattice gauge field. In contrast to earlier studies directly on the electronic degrees of freedom, we find evidence for a second-order deconfined quantum phase transition at which both the Néel and d -wave superconductivity orders vanish continuously. We compute correlators of the spinon-chargon composite with the same quantum numbers as the electron: we find a gapless Dirac dispersion inside the d -wave superconductor, turning into a gapped dispersion in the antiferromagnet.
Quantum fluctuations of the Néel state of the square lattice antiferromagnet are usually described by a CP 1 theory of bosonic spinons coupled to a U(1) gauge field, and with a global SU(2) spin rotation symmetry.Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge-2 Higgs fields, deconfined phases with Z 2 topological order possibly intertwined with discrete broken global symmetries.We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in π flux in each square lattice plaquette.Fluctuations about this π-flux state are described by (2 þ 1)-dimensional quantum chromodynamics (QCD 3 ) with a SU(2) gauge group and N f ¼ 2 flavors of massless Dirac fermions.It has recently been argued by Wang et al. [Deconfined Quantum Critical Points: Symmetries and Dualities, Phys. Rev. X 7, 031051 (2017).]that this QCD 3 theory describes the Néel-VBS quantum phase transition.We introduce adjoint Higgs fields in QCD 3 and obtain fermionic dual descriptions of the phases with Z 2 topological order obtained earlier using the bosonic CP 1 theory.We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state.The global phase diagram of these phases contains multicritical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.
We present a fermionic gauge theory for deconfined quantum criticality on the Shastry-Sutherland lattice and reveal its shared low-energy field-theoretic structure with the square lattice. Starting from an SU(2) πππ-flux parent state, we construct a continuum theory of Dirac spinons coupled to an SU(2) gauge field and adjoint Higgs fields whose condensates drive transitions to a staggered-flux U(1) spin liquid and a gapless Z2\mathbb{Z}_{2}Z2​ Dirac spin liquid. While the Shastry-Sutherland lattice permits additional symmetry-allowed fermion bilinears compared to the square lattice, the quantum field theories are identical up to additional irrelevant terms. Consequently, the Higgs potential structure and the leading low-energy theory coincide with the square-lattice case at the quantum critical point. The SO(5) critical point is expected to realize conformal deconfined criticality: we analyze it in a large flavor expansion, calculate its critical exponents, and identify the Yukawa coupling between the fermions and Higgs fields as the relevant perturbation that destabilizes it, consistent with pseudocritical behavior observed in recent Monte Carlo studies. We show that the emergent SO(5) order parameter acquires a large anomalous dimension at the critical point, leading to strongly enhanced Néel and VBS susceptibilities-a hallmark of fermionic deconfined quantum criticality consistent with numerical studies. Our results place recent numerical evidence for a gapless Z2\mathbb{Z}_{2}Z2​ Dirac spin liquid on the Shastry-Sutherland lattice within a controlled field-theoretic framework and demonstrate that fermionic deconfined criticality on the square lattice-including critical exponents and stability-extends to frustrated lattices with reduced symmetry.
We investigate the ground state phase diagram of an extended Hubbard model with a $\ensuremath{\pi}$-flux hopping term at half filling on a square lattice, with unbiased large-scale auxiliary-field quantum Monte Carlo simulations. As a function of interaction strength, there emerges an intermediate phase which realizes two interaction-driven quantum critical points, with the first between the Dirac semimetal and an insulating phase of weak valence bond solid (VBS) order, and the second separating the VBS order and an antiferromagnetic insulating phase. These intriguing quantum critical points are respectively bestowed with Gross-Neveu and deconfined quantum criticalities, and the critical exponents ${\ensuremath{\eta}}_{\text{VBS}}=0.6(1)$ and ${\ensuremath{\eta}}_{\text{AFM}}=0.58(3)$ at a deconfined quantum critical point satisfy the conformal field theory bootstrap bound. We also investigate the dynamical properties of the spin excitation and find the spin gap open near the first transition and closed at the second. The relevance of our findings in realizing deconfined quantum criticality in fermion systems and the implication to lattice models with further extended interactions such as those in quantum moir\'e systems, are discussed.
Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N ¼ 1, 2, and 3.The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm.In particular, one of the prominent phases for N > 1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z 2 gauge field in its deconfined phase.However, unlike a conventional Z 2 gauge theory, we do not impose "Gauss's Law" by hand; instead, it emerges because of spontaneous symmetry breaking.Correspondingly, unlike a conventional Z 2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature.By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N ¼ 2 and a valence-bond solid for N ¼ 3. Furthermore, for N ¼ 3, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.